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Generalized $F$-depth and graded nilpotent singularities

Kyle Maddox, Lance Edward Miller

Abstract

We address explicit constructions of new variants of $F$-nilpotent singularities. In particular, we explore how (generalized) weakly $F$-nilpotent singularities behave under gluing, Segre products, Veronese subrings, and the formation of diagonal hypersurface algebras. From these results, explicit examples are produced and we provide bounds on their Frobenius test exponents. To accomplish these tasks, we introduce the {\it generalized $F$-depth} in analogy to Lyubeznik's $F$-depth. These depth-like invariants track (generalized) weakly $F$-nilpotent singularities in a similar fashion as (generalized) depth tracks (generalized) Cohen-Macaulay singularities.

Generalized $F$-depth and graded nilpotent singularities

Abstract

We address explicit constructions of new variants of -nilpotent singularities. In particular, we explore how (generalized) weakly -nilpotent singularities behave under gluing, Segre products, Veronese subrings, and the formation of diagonal hypersurface algebras. From these results, explicit examples are produced and we provide bounds on their Frobenius test exponents. To accomplish these tasks, we introduce the {\it generalized -depth} in analogy to Lyubeznik's -depth. These depth-like invariants track (generalized) weakly -nilpotent singularities in a similar fashion as (generalized) depth tracks (generalized) Cohen-Macaulay singularities.

Paper Structure

This paper contains 22 sections, 35 theorems, 44 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Modules with Frobenius actions: the local case
  4. Tight and Frobenius closure of submodules
  5. Frobenius test exponents
  6. Nilpotent singularity types and depth-like invariants
  7. F-depth and generalized F-depth
  8. Gluing Nilpotent Singularities
  9. Gluing nilpotent singularity types
  10. Examples
  11. Frobenius test exponents for gluing
  12. Modules with Frobenius actions: the graded case
  13. Graded Frobenius actions
  14. Graded nilpotent singularities
  15. Segre Products
  16. ...and 7 more sections

Key Result

Theorem 1

(Corollary thm:GlueWeakFNil) Suppose $(R,\mathfrak{m})$ is a local ring of dimension $d \geq 2$ with ideals $\mathfrak{a}_1, \mathfrak{a}_2 \subset R$ such that $\mathfrak{a}_1 \cap \mathfrak{a}_2 = 0$. Assume $\dim R/\mathfrak{a}_1 = \dim R/\mathfrak{a}_2 = d$ and that $\dim R/(\mathfrak{a}_1 + \ma

Theorems & Definitions (104)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Definition 2.1
  • Remark 2.2
  • Example 2.3
  • Definition 2.4
  • Example 2.5
  • Remark 2.6
  • ...and 94 more